On the reachable set for the one-dimensional heat equation

International audienceThe goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x ∈ (−L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 > L any function which can be extended analytically on the square {x + iy, |x| + |y| ≤ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| < L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions

The Morozov's principle applied to data assimilation problems

This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation

Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography

The aim of electrical impedance tomography is to reconstruct the admittivity distribution inside a physical body from boundary measurements of current and voltage. Due to the severe ill-posedness of the underlying inverse problem, the functionality of impedance tomography relies heavily on accurate modelling of the measurement geometry. In particular, almost all reconstruction algorithms require the precise shape of the imaged body as an input. In this work, the need for prior geometric information is relaxed by introducing a Newton-type output least squares algorithm that reconstructs the admittivity distribution and the object shape simultaneously. The method is built in the framework of the complete electrode model and it is based on the Fr\'echet derivative of the corresponding current-to-voltage map with respect to the object boundary shape. The functionality of the technique is demonstrated via numerical experiments with simulated measurement data.Comment: 3 figure

Stability estimates for Navier-Stokes equations and application to inverse problems

International audienceIn this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method

Conditional stability for ill-posed elliptic Cauchy problems : the case of Lipschitz domains (part II)

This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with Lipschitz boundary. It completes the results obtained in \cite{bourgeois1} for domains of class $C^{1,1}$. This estimate is established by using an interior Carleman estimate and a technique based on a sequence of balls which approach the boundary. This technique is inspired from \cite{alessandrini}. We obtain a logarithmic stability estimate, the exponent of which is specified as a function of the boundary's singularity. Such stability estimate induces a convergence rate for the method of quasi-reversibility introduced in \cite{lions} to solve the Cauchy problems. The optimality of this convergence rate is tested numerically, precisely a discretized method of quasi-reversibility is performed by using a nonconforming finite element. The obtained results show very good agreement between theoretical and numerical convergence rates

The "exterior approach" applied to the inverse obstacle problem for the heat equation

International audienceIn this paper we consider the " exterior approach " to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our strategy of identification. 1. Introduction. This paper deals with the inverse obstacle problem for the heat equation, which can be described as follows. We consider a bounded domain D ⊂ R d , d ≥ 2, which contains an inclusion O. The temperature in the complementary domain Ω = D \ O satisfies the heat equation while the inclusion is characterized by a zero temperature. The inverse problem consists, from the knowledge of the lateral Cauchy data (that is both the temperature and the heat flux) on a subpart of the boundary ∂D during a certain interval of time (0, T) such that the temperature at time t = 0 is 0 in Ω, to identify the inclusion O. Such kind of inverse problem arises in thermal imaging, as briefly described in the introduction of [9]. The first attempts to solve such kind of problem numerically go back to the late 80's, as illustrated by [1], in which a least square method based on a shape derivative technique is used and numerical applications in 2D are presented. A shape derivative technique is also used in [11] in a 2D case as well, but the least square method is replaced by a Newton type method. Lastly, the shape derivative together with the least square method have recently been used in 3D cases [18]. The main feature of all these contributions is that they rely on the computation of forward problems in the domain Ω × (0, T): this computation obliges the authors to know one of the two lateral Cauchy data (either the temperature or the heat flux) on the whole boundary ∂D of D. In [20], the authors introduce the so-called " enclosure method " , which enables them to recover an approximation of the convex hull of the inclusion without computing any forward problem. Note however that the lateral Cauchy data has to be known on the whole boundary ∂D. The present paper concerns the " exterior approach " , which is an alternative method to solve the inverse obstacle problem. Like in [20], it does not need to compute the solution of the forward problem and in addition, it is applicable even if the lateral Cauchy data are known only on a subpart of ∂D, while no data are given on the complementary part. The " exterior approach " consists in defining a sequence of domains that converges in a certain sense to the inclusion we are looking for. More precisely, the nth step consists, 1. for a given inclusion O n , in approximating the temperature in Ω n × (0, T) (Ω n := D \ O n) with the help of a quasi-reversibility method, 2. for a given temperature in Ω n × (0, T), in computing an updated inclusion O n+1 with the help of a level set method. Such " exterior approach " has already been successfully used to solve inverse obstacle problems for the Laplace equation [5, 4, 15] and for the Stokes system [6]. It has also been used for the heat equation in the 1D case [2]: the problem in this simple case might be considered as a toy problem since the inclusion reduces to a point in some bounded interval. The objective of the present paper is to extend the " exterior approach " for the heat equation to any dimension of space, with numerical applications in the 2D case. Let us shed some light on the two steps o

ITERATED QUASI-REVERSIBILITY METHOD APPLIED TO ELLIPTIC AND PARABOLIC DATA COMPLETION PROBLEMS

International audienceWe study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method prove to be efficient even with highly corrupted data

Méthodes de quasi-réversibilité et de lignes de niveau appliquées aux problèmes inverses elliptiques.

This work considers the quasi-reversibility method for solving some inverse problems, a typical example being the inverse obstacle problem. We propose for the latter a new approach that couples the quasi-reversibility method and a level set method. More precisely, from a candidate open domain C, we solve a Cauchy problem outside C, and then update C using the level set method. The approximated solution of the Cauchy problem is obtained by using the quasi-reversibility method introduced by J.L. Lions and R. Lattès in the sixties. We propose different formulations of this method, as well as its discretization by nonconforming finite elements adapted to the framework of Sobolev space H2, and we prove the convergence of the finite elements. In the presence of noisy data, we introduce a new method based on duality in optimization and the Morozov's discrepancy principle. We establish a relationship between this method and the quasi-reversibility; in particular, we show that it provides regularized data and a relevant regularization parameter that increase the efficiency of quasi-reversibility. Regarding the update of the open domain C, we propose two very different level set approaches: one is based on an eikonal equation, the other on a Poisson equation. We prove that this two approaches guarantee convergence to the obstacle. Finally, we present numerical results for this coupled quasi-reversibility/level set approach in different situations: inverse obstacle problem with Dirichlet condition, detection of some defects in an elastoplastic structure...Ce travail s'intéresse à l'utilisation de la méthode de quasi-réversibilité pour la résolution de problèmes inverses, un exemple typique étant le problème inverse de l'obstacle. Nous proposons pour ce dernier une nouvelle approche couplant la méthode de quasi-réversibilité et une méthode de lignes de niveau. Plus précisément, à partir d'un ouvert candidat C, nous résolvons un problème de Cauchy à l'extérieur de C, puis nous mettons à jour cet ouvert par la méthode de lignes de niveau. La solution approchée du problème de Cauchy est obtenue en utilisant la méthode de quasi-réversibilité, introduite par J.L. Lions et R. Lattès dans les années soixante. Nous proposons différentes formulations de cette méthode, ainsi que sa discrétisation par éléments finis non conformes adaptés à l'espace de Sobolev H2, et nous prouvons la convergence des éléments finis. En présence d'une donnée bruitée, nous introduisons une nouvelle méthode basée sur la dualité en optimisation et le principe de Morozov. Nous montrons que cette méthode fournit des données régularisées et un choix de paramètre de régularisation pertinent pour la quasi-réversibilité. En ce qui concerne la mise à jour de l'ouvert C, nous proposons deux méthodes de lignes de niveau très différentes : la première est basée sur une équation eikonale, la seconde sur une équation de Poisson. Nous prouvons que ces deux approches assurent la convergence vers l'obstacle. Finalement, nous présentons des résultats numériques pour cette approche couplant quasi-réversibilité/lignes de niveau dans différentes situations : problème inverse de l'obstacle avec condition de Dirichlet, détection de défauts dans une structure élasto-plastique..

A dual approach to Kohn-Vogelius regularization applied to data completion problem

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Backward uniqueness results for some parabolic equations in an infinite rod

International audienceThe goal of this article is to provide backward uniqueness results for several models of parabolic equations set on the half line, namely the heat equation, and the heat equation with quadratic potential and with purely imaginary quadratic potentials, with non-homogeneous boundary conditions. Such result can thus also be interpreted as a strong lack of controllability on the half line, as it shows that only the trivial initial datum can be steered to zero. Our results are based on the explicit knowledge of the kernel of each equation, and standard arguments from complex analysis, namely the Phragmén Lindelöf principle